Theorems · Theorem · linear algebra
LinearMap.injective_iff_surjective
∀ {K : Type u} {V : Type v} [inst : DivisionRing K] [inst_1 : AddCommGroup V] [inst_2 : Module K V]
[FiniteDimensional K V] {f : V →ₗ[K] V}, Function.Injective ⇑f ↔ Function.Surjective ⇑fOn a finite-dimensional space, a linear map is injective if and only if it is surjective.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement and proof · cited by 10,215
- FiniteDimensionalstatement and proof · cited by 1,854
- LinearMap.compproof · cited by 1,642
- DivisionRingstatement and proof · cited by 1,062
- LinearMap.idproof · cited by 625
- LinearMap.range_eq_topproof · cited by 107
- LinearMap.ext_iffproof · cited by 32
- Function.RightInverse.injectiveproof · cited by 20
Cited by3
Results whose statement or proof uses this declaration.
- AlgHom.bijectiveproof · cited by 2
- LinearMap.ker_eq_bot_iff_range_eq_topproof · cited by 1
- FiniteDimensional.exists_mul_eq_oneproof · cited by 1